We can study the bulk spectral properties of the quasiperiodic model along a so-called approximant path, where we explore finite periodic models that capture well the incommensurate length scales competition within their enlarged unit cell [1]}, [2]}. Beyond the finite unit cell length scale, we use a periodic approximation and apply Bloch’s theorem. We recall that when the approximant length scale in real space is equivalent to \(q\) , the corresponding first Brillouin zone in momentum space is \(q\)  times smaller than for a simple periodic lattice. As a consequence, the energy dispersion \(E(k)\)  of the periodic chain folds from the larger to the reduced 1st Brillouin zone, thus forming \(q\)  different energy bands. Note that the bands may overlap depending on the specific model, e.g., if a symmetry persists under the folding. As an example for such an approximant approach, we consider the interpolating quasiperiodic function (REF ), see Fig. REF . The irrational modulation \(\tau \)  has a unique continued fraction representation
\(\frac{1}{\tau }=a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{ \ddots + \cfrac{1}{a_n} }}}\,,\) 
